In some studies with bivariate left-censored data, the underlying response variables also attain a zero value with a positive discrete probability. We introduced parametric and semi-parametric regression models for these bivariate zero-inflated left-censored survival data. The different parameters in the model are estimated using maximum likelihood techniques. The numerical optimization of the likelihood becomes more difficult as the number of parameters increases. Fortunately, the model structure suggests that a two-stage estimation procedure can be considered. Firstly we estimate the parameters in the margins, ignoring the dependence of the two components. The second stage involves maximum likelihood of the dependence parameters with the univariate parameters held fixed from the first stage. We derived a partitioned form for the asymptotic variance-covariance matrix of the two-stage parametric estimators and discussed a jackknife estimator for this matrix. In the simulation study, we showed that the two-stage parametric and semi-parametric estimation methods perform well, especially when the association between two non-zero responses is low or moderate. Finally, we have applied our regression model on a practical data set of ethanolinduced sleep time in mice.